Binomial Treesintermediate

Delta

Delta ( $\Delta$ ) is the sensitivity of an option price to a small change in the underlying. In a one-step binomial model, $\Delta = (f_u - f_d)/(S_u - S_d)$ , which is also the hedge ratio, the number of shares to hold per option to make the portfolio locally riskless. Call deltas lie in $[0, 1]$ , put deltas in $[-1, 0]$ , and both deltas change as the stock moves, a second-order effect captured by gamma.

Why it matters

Delta is the slope of the option-price curve plotted against the stock price. Deep in-the-money calls behave almost like the stock itself, so $\Delta \approx 1$ . Deep out-of-the-money calls barely react, so $\Delta \approx 0$ . At-the-money calls sit around $\Delta \approx 0.5$ . The same number that tells you how the price reacts also tells you how many shares to hold to neutralise that reaction. That dual role is why delta is the first Greek every options trader learns.

Formulas

Binomial delta

\Delta = \frac{f_u - f_d}{S_u - S_d}

Equal to the number of shares in the replicating portfolio for one option (Hull 2022, §13.1).

Black–Scholes call delta

\Delta_{\text{call}} = N(d_1)

Continuous-time limit. Always between 0 and 1.

Approximate price change

\Delta f \approx \Delta \cdot \Delta S

Worked examples

Scenario

One-step tree with $S_0 =$ A$100, $u = 1.1$ , $d = 0.9$ , so $S_u =$ A$110 and $S_d =$ A$90. European call struck at $K =$ A$100 has $f_u =$ A$10 and $f_d = 0$ .

Solution

$\Delta = (10 - 0)/(110 - 90) = 0.5$ . To hedge one short call, hold 0.5 shares. If the stock moves up to A$110, the share position gains A$5 and the short call loses A$10, leaving a net loss of A$5 to be funded by the borrowed cash leg of the replicating portfolio. The same logic in reverse applies to the down move. The portfolio's payoff is locally risk-free.

Common mistakes

✗Delta is constant. It is not. Delta moves with $S$ , $T$ , and $\sigma$ , and the rate of change is gamma $\Gamma = \partial \Delta / \partial S$ . A delta-hedged book stays neutral only locally and must be rebalanced as the stock moves.
✗Delta equals the probability of finishing in the money. Close but not equal. For a European call, $\Delta = N(d_1)$ , while the risk-neutral probability of finishing ITM is $N(d_2)$ . The difference reflects that delta weighs the in-the-money states by their stock prices, not just their probabilities (Hull 2022, §17.4).

Revision bullets

•Binomial delta $\Delta = (f_u - f_d)/(S_u - S_d)$
•Call delta in $[0, 1]$ , put delta in $[-1, 0]$
•Also the hedge ratio in the replicating portfolio
•Not constant, changes via gamma effect
• $N(d_1)$ in the Black–Scholes continuous-time limit

Quick check

If a call option has $\Delta = 0.7$ , a A$2 rise in the stock price changes the option price by approximately:

Connected topics

1-Step Tree Delta Hedge BSM Intro

In learning paths

Pricing Models Path

Sources

Hull (2022), §13.1
Hull, John C. Options, Futures, and Other Derivatives. 11th ed. Pearson, 2022. ISBN 978-0-13-693997-9.
Derives the binomial delta as the number of shares in the no-arbitrage replicating portfolio and links it to the continuous-time Greek.
Black & Scholes (1973)
Black, Fischer and Myron Scholes. "The Pricing of Options and Corporate Liabilities." Journal of Political Economy 81(3), 1973, pp. 637–654.
Original derivation of the dynamic hedging argument in which delta is the share weight that eliminates first-order stock risk.
Cox, Ross & Rubinstein (1979)
Cox, John C., Stephen A. Ross and Mark Rubinstein. "Option Pricing: A Simplified Approach." Journal of Financial Economics 7(3), 1979, pp. 229–263.
Discrete-time derivation that exposes delta as the simple difference quotient $(f_u - f_d)/(S_u - S_d)$.

How to cite this page

Dr. Phil's Quant Lab. (2026). Delta. Derivatives Atlas. https://phucnguyenvan.com/concept/delta

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